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Theorem monpropd 15653
Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same monomorphisms. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
monpropd.3  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
monpropd.4  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
monpropd.c  |-  ( ph  ->  C  e.  Cat )
monpropd.d  |-  ( ph  ->  D  e.  Cat )
Assertion
Ref Expression
monpropd  |-  ( ph  ->  (Mono `  C )  =  (Mono `  D )
)

Proof of Theorem monpropd
Dummy variables  a 
b  c  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2452 . . . . . . . . . . . 12  |-  ( Base `  C )  =  (
Base `  C )
2 eqid 2452 . . . . . . . . . . . 12  |-  ( Hom  `  C )  =  ( Hom  `  C )
3 eqid 2452 . . . . . . . . . . . 12  |-  ( Hom  `  D )  =  ( Hom  `  D )
4 monpropd.3 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
54ad2antrr 737 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
65ad2antrr 737 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  a  e.  ( Base `  C ) )  /\  b  e.  ( Base `  C ) )  /\  f  e.  ( a
( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C ) )  -> 
( Hom f  `  C )  =  ( Hom f  `  D ) )
7 simpr 467 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  a  e.  ( Base `  C ) )  /\  b  e.  ( Base `  C ) )  /\  f  e.  ( a
( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C ) )  -> 
c  e.  ( Base `  C ) )
8 simp-4r 782 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  a  e.  ( Base `  C ) )  /\  b  e.  ( Base `  C ) )  /\  f  e.  ( a
( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C ) )  -> 
a  e.  ( Base `  C ) )
91, 2, 3, 6, 7, 8homfeqval 15613 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  a  e.  ( Base `  C ) )  /\  b  e.  ( Base `  C ) )  /\  f  e.  ( a
( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C ) )  -> 
( c ( Hom  `  C ) a )  =  ( c ( Hom  `  D )
a ) )
10 eqid 2452 . . . . . . . . . . . 12  |-  (comp `  C )  =  (comp `  C )
11 eqid 2452 . . . . . . . . . . . 12  |-  (comp `  D )  =  (comp `  D )
124ad5antr 745 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  a  e.  ( Base `  C )
)  /\  b  e.  ( Base `  C )
)  /\  f  e.  ( a ( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C
) )  /\  g  e.  ( c ( Hom  `  C ) a ) )  ->  ( Hom f  `  C
)  =  ( Hom f  `  D ) )
13 monpropd.4 . . . . . . . . . . . . 13  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
1413ad5antr 745 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  a  e.  ( Base `  C )
)  /\  b  e.  ( Base `  C )
)  /\  f  e.  ( a ( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C
) )  /\  g  e.  ( c ( Hom  `  C ) a ) )  ->  (compf `  C )  =  (compf `  D ) )
15 simplr 767 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  a  e.  ( Base `  C )
)  /\  b  e.  ( Base `  C )
)  /\  f  e.  ( a ( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C
) )  /\  g  e.  ( c ( Hom  `  C ) a ) )  ->  c  e.  ( Base `  C )
)
16 simp-5r 784 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  a  e.  ( Base `  C )
)  /\  b  e.  ( Base `  C )
)  /\  f  e.  ( a ( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C
) )  /\  g  e.  ( c ( Hom  `  C ) a ) )  ->  a  e.  ( Base `  C )
)
17 simp-4r 782 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  a  e.  ( Base `  C )
)  /\  b  e.  ( Base `  C )
)  /\  f  e.  ( a ( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C
) )  /\  g  e.  ( c ( Hom  `  C ) a ) )  ->  b  e.  ( Base `  C )
)
18 simpr 467 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  a  e.  ( Base `  C )
)  /\  b  e.  ( Base `  C )
)  /\  f  e.  ( a ( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C
) )  /\  g  e.  ( c ( Hom  `  C ) a ) )  ->  g  e.  ( c ( Hom  `  C ) a ) )
19 simpllr 774 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  a  e.  ( Base `  C )
)  /\  b  e.  ( Base `  C )
)  /\  f  e.  ( a ( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C
) )  /\  g  e.  ( c ( Hom  `  C ) a ) )  ->  f  e.  ( a ( Hom  `  C ) b ) )
201, 2, 10, 11, 12, 14, 15, 16, 17, 18, 19comfeqval 15624 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  a  e.  ( Base `  C )
)  /\  b  e.  ( Base `  C )
)  /\  f  e.  ( a ( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C
) )  /\  g  e.  ( c ( Hom  `  C ) a ) )  ->  ( f
( <. c ,  a
>. (comp `  C )
b ) g )  =  ( f (
<. c ,  a >.
(comp `  D )
b ) g ) )
219, 20mpteq12dva 4452 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  a  e.  ( Base `  C ) )  /\  b  e.  ( Base `  C ) )  /\  f  e.  ( a
( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C ) )  -> 
( g  e.  ( c ( Hom  `  C
) a )  |->  ( f ( <. c ,  a >. (comp `  C ) b ) g ) )  =  ( g  e.  ( c ( Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) )
2221cnveqd 4988 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  a  e.  ( Base `  C ) )  /\  b  e.  ( Base `  C ) )  /\  f  e.  ( a
( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C ) )  ->  `' ( g  e.  ( c ( Hom  `  C ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  C )
b ) g ) )  =  `' ( g  e.  ( c ( Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) )
2322funeqd 5582 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  a  e.  ( Base `  C ) )  /\  b  e.  ( Base `  C ) )  /\  f  e.  ( a
( Hom  `  C ) b ) )  /\  c  e.  ( Base `  C ) )  -> 
( Fun  `' (
g  e.  ( c ( Hom  `  C
) a )  |->  ( f ( <. c ,  a >. (comp `  C ) b ) g ) )  <->  Fun  `' ( g  e.  ( c ( Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) ) )
2423ralbidva 2809 . . . . . . 7  |-  ( ( ( ( ph  /\  a  e.  ( Base `  C ) )  /\  b  e.  ( Base `  C ) )  /\  f  e.  ( a
( Hom  `  C ) b ) )  -> 
( A. c  e.  ( Base `  C
) Fun  `' (
g  e.  ( c ( Hom  `  C
) a )  |->  ( f ( <. c ,  a >. (comp `  C ) b ) g ) )  <->  A. c  e.  ( Base `  C
) Fun  `' (
g  e.  ( c ( Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) ) )
2524rabbidva 3003 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  { f  e.  ( a ( Hom  `  C )
b )  |  A. c  e.  ( Base `  C ) Fun  `' ( g  e.  ( c ( Hom  `  C
) a )  |->  ( f ( <. c ,  a >. (comp `  C ) b ) g ) ) }  =  { f  e.  ( a ( Hom  `  C ) b )  |  A. c  e.  ( Base `  C
) Fun  `' (
g  e.  ( c ( Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) } )
26 simplr 767 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  a  e.  ( Base `  C
) )
27 simpr 467 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  b  e.  ( Base `  C
) )
281, 2, 3, 5, 26, 27homfeqval 15613 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  (
a ( Hom  `  C
) b )  =  ( a ( Hom  `  D ) b ) )
294homfeqbas 15612 . . . . . . . . 9  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
3029ad2antrr 737 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  ( Base `  C )  =  ( Base `  D
) )
3130raleqdv 2961 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  ( A. c  e.  ( Base `  C ) Fun  `' ( g  e.  ( c ( Hom  `  D ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  D )
b ) g ) )  <->  A. c  e.  (
Base `  D ) Fun  `' ( g  e.  ( c ( Hom  `  D ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  D )
b ) g ) ) ) )
3228, 31rabeqbidv 3008 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  { f  e.  ( a ( Hom  `  C )
b )  |  A. c  e.  ( Base `  C ) Fun  `' ( g  e.  ( c ( Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) }  =  { f  e.  ( a ( Hom  `  D ) b )  |  A. c  e.  ( Base `  D
) Fun  `' (
g  e.  ( c ( Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) } )
3325, 32eqtrd 2486 . . . . 5  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  { f  e.  ( a ( Hom  `  C )
b )  |  A. c  e.  ( Base `  C ) Fun  `' ( g  e.  ( c ( Hom  `  C
) a )  |->  ( f ( <. c ,  a >. (comp `  C ) b ) g ) ) }  =  { f  e.  ( a ( Hom  `  D ) b )  |  A. c  e.  ( Base `  D
) Fun  `' (
g  e.  ( c ( Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) } )
34333impa 1205 . . . 4  |-  ( (
ph  /\  a  e.  ( Base `  C )  /\  b  e.  ( Base `  C ) )  ->  { f  e.  ( a ( Hom  `  C ) b )  |  A. c  e.  ( Base `  C
) Fun  `' (
g  e.  ( c ( Hom  `  C
) a )  |->  ( f ( <. c ,  a >. (comp `  C ) b ) g ) ) }  =  { f  e.  ( a ( Hom  `  D ) b )  |  A. c  e.  ( Base `  D
) Fun  `' (
g  e.  ( c ( Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) } )
3534mpt2eq3dva 6343 . . 3  |-  ( ph  ->  ( a  e.  (
Base `  C ) ,  b  e.  ( Base `  C )  |->  { f  e.  ( a ( Hom  `  C
) b )  | 
A. c  e.  (
Base `  C ) Fun  `' ( g  e.  ( c ( Hom  `  C ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  C )
b ) g ) ) } )  =  ( a  e.  (
Base `  C ) ,  b  e.  ( Base `  C )  |->  { f  e.  ( a ( Hom  `  D
) b )  | 
A. c  e.  (
Base `  D ) Fun  `' ( g  e.  ( c ( Hom  `  D ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  D )
b ) g ) ) } ) )
36 mpt2eq12 6339 . . . 4  |-  ( ( ( Base `  C
)  =  ( Base `  D )  /\  ( Base `  C )  =  ( Base `  D
) )  ->  (
a  e.  ( Base `  C ) ,  b  e.  ( Base `  C
)  |->  { f  e.  ( a ( Hom  `  D ) b )  |  A. c  e.  ( Base `  D
) Fun  `' (
g  e.  ( c ( Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) } )  =  ( a  e.  ( Base `  D
) ,  b  e.  ( Base `  D
)  |->  { f  e.  ( a ( Hom  `  D ) b )  |  A. c  e.  ( Base `  D
) Fun  `' (
g  e.  ( c ( Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) } ) )
3729, 29, 36syl2anc 671 . . 3  |-  ( ph  ->  ( a  e.  (
Base `  C ) ,  b  e.  ( Base `  C )  |->  { f  e.  ( a ( Hom  `  D
) b )  | 
A. c  e.  (
Base `  D ) Fun  `' ( g  e.  ( c ( Hom  `  D ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  D )
b ) g ) ) } )  =  ( a  e.  (
Base `  D ) ,  b  e.  ( Base `  D )  |->  { f  e.  ( a ( Hom  `  D
) b )  | 
A. c  e.  (
Base `  D ) Fun  `' ( g  e.  ( c ( Hom  `  D ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  D )
b ) g ) ) } ) )
3835, 37eqtrd 2486 . 2  |-  ( ph  ->  ( a  e.  (
Base `  C ) ,  b  e.  ( Base `  C )  |->  { f  e.  ( a ( Hom  `  C
) b )  | 
A. c  e.  (
Base `  C ) Fun  `' ( g  e.  ( c ( Hom  `  C ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  C )
b ) g ) ) } )  =  ( a  e.  (
Base `  D ) ,  b  e.  ( Base `  D )  |->  { f  e.  ( a ( Hom  `  D
) b )  | 
A. c  e.  (
Base `  D ) Fun  `' ( g  e.  ( c ( Hom  `  D ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  D )
b ) g ) ) } ) )
39 eqid 2452 . . 3  |-  (Mono `  C )  =  (Mono `  C )
40 monpropd.c . . 3  |-  ( ph  ->  C  e.  Cat )
411, 2, 10, 39, 40monfval 15648 . 2  |-  ( ph  ->  (Mono `  C )  =  ( a  e.  ( Base `  C
) ,  b  e.  ( Base `  C
)  |->  { f  e.  ( a ( Hom  `  C ) b )  |  A. c  e.  ( Base `  C
) Fun  `' (
g  e.  ( c ( Hom  `  C
) a )  |->  ( f ( <. c ,  a >. (comp `  C ) b ) g ) ) } ) )
42 eqid 2452 . . 3  |-  ( Base `  D )  =  (
Base `  D )
43 eqid 2452 . . 3  |-  (Mono `  D )  =  (Mono `  D )
44 monpropd.d . . 3  |-  ( ph  ->  D  e.  Cat )
4542, 3, 11, 43, 44monfval 15648 . 2  |-  ( ph  ->  (Mono `  D )  =  ( a  e.  ( Base `  D
) ,  b  e.  ( Base `  D
)  |->  { f  e.  ( a ( Hom  `  D ) b )  |  A. c  e.  ( Base `  D
) Fun  `' (
g  e.  ( c ( Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) } ) )
4638, 41, 453eqtr4d 2496 1  |-  ( ph  ->  (Mono `  C )  =  (Mono `  D )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1448    e. wcel 1891   A.wral 2737   {crab 2741   <.cop 3942    |-> cmpt 4433   `'ccnv 4811   Fun wfun 5555   ` cfv 5561  (class class class)co 6276    |-> cmpt2 6278   Basecbs 15132   Hom chom 15212  compcco 15213   Catccat 15581   Hom f chomf 15583  compfccomf 15584  Monocmon 15644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-rep 4487  ax-sep 4497  ax-nul 4506  ax-pow 4554  ax-pr 4612  ax-un 6571
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3015  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4169  df-iun 4250  df-br 4375  df-opab 4434  df-mpt 4435  df-id 4727  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5525  df-fun 5563  df-fn 5564  df-f 5565  df-f1 5566  df-fo 5567  df-f1o 5568  df-fv 5569  df-ov 6279  df-oprab 6280  df-mpt2 6281  df-1st 6781  df-2nd 6782  df-homf 15587  df-comf 15588  df-mon 15646
This theorem is referenced by:  oppcepi  15655
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